Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D

Percentage Accurate: 99.8% → 99.9%

Time: 7.2s

Alternatives: 6

Speedup: 1.4×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))

C

double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * ((((x * 3.0d0) * x) - (x * 4.0d0)) + 1.0d0)
end function

Java

public static double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Python

def code(x):
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0)

Julia

function code(x)
	return Float64(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
end

Wolfram

code[x_] := N[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))

C

double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * ((((x * 3.0d0) * x) - (x * 4.0d0)) + 1.0d0)
end function

Java

public static double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Python

def code(x):
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0)

Julia

function code(x)
	return Float64(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
end

Wolfram

code[x_] := N[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 3 + x \cdot \left(-12 + x \cdot 9\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (* x (+ -12.0 (* x 9.0)))))

C

double code(double x) {
	return 3.0 + (x * (-12.0 + (x * 9.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + (x * ((-12.0d0) + (x * 9.0d0)))
end function

Java

public static double code(double x) {
	return 3.0 + (x * (-12.0 + (x * 9.0)));
}

Python

def code(x):
	return 3.0 + (x * (-12.0 + (x * 9.0)))

Julia

function code(x)
	return Float64(3.0 + Float64(x * Float64(-12.0 + Float64(x * 9.0))))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 + (x * (-12.0 + (x * 9.0)));
end

Wolfram

code[x_] := N[(3.0 + N[(x * N[(-12.0 + N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Step-by-step derivation

   1. associate-+l- 99.4%

      \[\leadsto 3 \cdot \color{blue}{\left(\left(x \cdot 3\right) \cdot x - \left(x \cdot 4 - 1\right)\right)} \]

   2. associate-*l* 99.4%

      \[\leadsto 3 \cdot \left(\color{blue}{x \cdot \left(3 \cdot x\right)} - \left(x \cdot 4 - 1\right)\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{3 \cdot \left(x \cdot \left(3 \cdot x\right) - \left(x \cdot 4 - 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.4%

   \[\leadsto 3 \cdot \color{blue}{\left(1 + \left(-4 \cdot x + 3 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 99.4%

      \[\leadsto 3 \cdot \left(1 + \color{blue}{\left(3 \cdot {x}^{2} + -4 \cdot x\right)}\right) \]

   2. fma-def 99.4%

      \[\leadsto 3 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(3, {x}^{2}, -4 \cdot x\right)}\right) \]

7. Simplified 99.4%

   \[\leadsto 3 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(3, {x}^{2}, -4 \cdot x\right)\right)} \]

8. Taylor expanded in x around 0 99.5%

   \[\leadsto \color{blue}{3 + \left(-12 \cdot x + 9 \cdot {x}^{2}\right)} \]
9. Step-by-step derivation

   1. unpow2 99.5%

      \[\leadsto 3 + \left(-12 \cdot x + 9 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   2. associate-*r* 99.5%

      \[\leadsto 3 + \left(-12 \cdot x + \color{blue}{\left(9 \cdot x\right) \cdot x}\right) \]

   3. *-commutative 99.5%

      \[\leadsto 3 + \left(-12 \cdot x + \color{blue}{\left(x \cdot 9\right)} \cdot x\right) \]

   4. distribute-rgt-out 99.9%

      \[\leadsto 3 + \color{blue}{x \cdot \left(-12 + x \cdot 9\right)} \]

10. Simplified 99.9%

    \[\leadsto \color{blue}{3 + x \cdot \left(-12 + x \cdot 9\right)} \]

11. Final simplification 99.9%

    \[\leadsto 3 + x \cdot \left(-12 + x \cdot 9\right) \]
12. Add Preprocessing

Alternative 2: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.58\right):\\ \;\;\;\;x \cdot \left(-12 + x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 0.58)))
   (* x (+ -12.0 (* x 9.0)))
   (+ 3.0 (* x -12.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.58)) {
		tmp = x * (-12.0 + (x * 9.0));
	} else {
		tmp = 3.0 + (x * -12.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.58d0)) .or. (.not. (x <= 0.58d0))) then
        tmp = x * ((-12.0d0) + (x * 9.0d0))
    else
        tmp = 3.0d0 + (x * (-12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.58)) {
		tmp = x * (-12.0 + (x * 9.0));
	} else {
		tmp = 3.0 + (x * -12.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) or not (x <= 0.58):
		tmp = x * (-12.0 + (x * 9.0))
	else:
		tmp = 3.0 + (x * -12.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.58) || !(x <= 0.58))
		tmp = Float64(x * Float64(-12.0 + Float64(x * 9.0)));
	else
		tmp = Float64(3.0 + Float64(x * -12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.58) || ~((x <= 0.58)))
		tmp = x * (-12.0 + (x * 9.0));
	else
		tmp = 3.0 + (x * -12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.58], N[Not[LessEqual[x, 0.58]], $MachinePrecision]], N[(x * N[(-12.0 + N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996 or 0.57999999999999996 < x

   1. Initial program 98.9%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 98.9%

         \[\leadsto 3 \cdot \color{blue}{\left(\left(x \cdot 3\right) \cdot x - \left(x \cdot 4 - 1\right)\right)} \]

      2. associate-*l* 98.9%

         \[\leadsto 3 \cdot \left(\color{blue}{x \cdot \left(3 \cdot x\right)} - \left(x \cdot 4 - 1\right)\right) \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \left(3 \cdot x\right) - \left(x \cdot 4 - 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.9%

      \[\leadsto 3 \cdot \color{blue}{\left(1 + \left(-4 \cdot x + 3 \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 98.9%

         \[\leadsto 3 \cdot \left(1 + \color{blue}{\left(3 \cdot {x}^{2} + -4 \cdot x\right)}\right) \]

      2. fma-def 98.9%

         \[\leadsto 3 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(3, {x}^{2}, -4 \cdot x\right)}\right) \]

   7. Simplified 98.9%

      \[\leadsto 3 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(3, {x}^{2}, -4 \cdot x\right)\right)} \]

   8. Taylor expanded in x around inf 97.8%

      \[\leadsto \color{blue}{-12 \cdot x + 9 \cdot {x}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 97.8%

         \[\leadsto -12 \cdot x + 9 \cdot \color{blue}{\left(x \cdot x\right)} \]

      2. associate-*r* 97.8%

         \[\leadsto -12 \cdot x + \color{blue}{\left(9 \cdot x\right) \cdot x} \]

      3. *-commutative 97.8%

         \[\leadsto -12 \cdot x + \color{blue}{\left(x \cdot 9\right)} \cdot x \]

      4. distribute-rgt-out 98.6%

         \[\leadsto \color{blue}{x \cdot \left(-12 + x \cdot 9\right)} \]

   10. Simplified 98.6%

       \[\leadsto \color{blue}{x \cdot \left(-12 + x \cdot 9\right)} \]

   if -0.57999999999999996 < x < 0.57999999999999996

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto 3 \cdot \color{blue}{\left(\left(x \cdot 3\right) \cdot x - \left(x \cdot 4 - 1\right)\right)} \]

      2. associate-*l* 100.0%

         \[\leadsto 3 \cdot \left(\color{blue}{x \cdot \left(3 \cdot x\right)} - \left(x \cdot 4 - 1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \left(3 \cdot x\right) - \left(x \cdot 4 - 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.4%

      \[\leadsto \color{blue}{3 + -12 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto 3 + \color{blue}{x \cdot -12} \]

   7. Simplified 98.4%

      \[\leadsto \color{blue}{3 + x \cdot -12} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.58\right):\\ \;\;\;\;x \cdot \left(-12 + x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 0.2))) (* x (* x 9.0)) 3.0))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.2)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.58d0)) .or. (.not. (x <= 0.2d0))) then
        tmp = x * (x * 9.0d0)
    else
        tmp = 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.2)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) or not (x <= 0.2):
		tmp = x * (x * 9.0)
	else:
		tmp = 3.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.58) || !(x <= 0.2))
		tmp = Float64(x * Float64(x * 9.0));
	else
		tmp = 3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.58) || ~((x <= 0.2)))
		tmp = x * (x * 9.0);
	else
		tmp = 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.58], N[Not[LessEqual[x, 0.2]], $MachinePrecision]], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], 3.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996 or 0.20000000000000001 < x

   1. Initial program 98.9%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 98.9%

         \[\leadsto 3 \cdot \color{blue}{\left(\left(x \cdot 3\right) \cdot x - \left(x \cdot 4 - 1\right)\right)} \]

      2. associate-*l* 98.9%

         \[\leadsto 3 \cdot \left(\color{blue}{x \cdot \left(3 \cdot x\right)} - \left(x \cdot 4 - 1\right)\right) \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \left(3 \cdot x\right) - \left(x \cdot 4 - 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.9%

      \[\leadsto 3 \cdot \color{blue}{\left(1 + \left(-4 \cdot x + 3 \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 98.9%

         \[\leadsto 3 \cdot \left(1 + \color{blue}{\left(3 \cdot {x}^{2} + -4 \cdot x\right)}\right) \]

      2. fma-def 98.9%

         \[\leadsto 3 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(3, {x}^{2}, -4 \cdot x\right)}\right) \]

   7. Simplified 98.9%

      \[\leadsto 3 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(3, {x}^{2}, -4 \cdot x\right)\right)} \]

   8. Taylor expanded in x around inf 97.8%

      \[\leadsto \color{blue}{-12 \cdot x + 9 \cdot {x}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 97.8%

         \[\leadsto -12 \cdot x + 9 \cdot \color{blue}{\left(x \cdot x\right)} \]

      2. associate-*r* 97.8%

         \[\leadsto -12 \cdot x + \color{blue}{\left(9 \cdot x\right) \cdot x} \]

      3. *-commutative 97.8%

         \[\leadsto -12 \cdot x + \color{blue}{\left(x \cdot 9\right)} \cdot x \]

      4. distribute-rgt-out 98.6%

         \[\leadsto \color{blue}{x \cdot \left(-12 + x \cdot 9\right)} \]

   10. Simplified 98.6%

       \[\leadsto \color{blue}{x \cdot \left(-12 + x \cdot 9\right)} \]

   11. Taylor expanded in x around inf 97.7%

       \[\leadsto x \cdot \color{blue}{\left(9 \cdot x\right)} \]
   12. Step-by-step derivation

       1. *-commutative 97.7%

          \[\leadsto x \cdot \color{blue}{\left(x \cdot 9\right)} \]

   13. Simplified 97.7%

       \[\leadsto x \cdot \color{blue}{\left(x \cdot 9\right)} \]

   if -0.57999999999999996 < x < 0.20000000000000001

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto 3 \cdot \color{blue}{\left(\left(x \cdot 3\right) \cdot x - \left(x \cdot 4 - 1\right)\right)} \]

      2. associate-*l* 100.0%

         \[\leadsto 3 \cdot \left(\color{blue}{x \cdot \left(3 \cdot x\right)} - \left(x \cdot 4 - 1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \left(3 \cdot x\right) - \left(x \cdot 4 - 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 97.3%

      \[\leadsto \color{blue}{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 0.215\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 0.215))) (* x (* x 9.0)) (+ 3.0 (* x -12.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 0.215)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0 + (x * -12.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 0.215d0))) then
        tmp = x * (x * 9.0d0)
    else
        tmp = 3.0d0 + (x * (-12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 0.215)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0 + (x * -12.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.55) or not (x <= 0.215):
		tmp = x * (x * 9.0)
	else:
		tmp = 3.0 + (x * -12.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 0.215))
		tmp = Float64(x * Float64(x * 9.0));
	else
		tmp = Float64(3.0 + Float64(x * -12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 0.215)))
		tmp = x * (x * 9.0);
	else
		tmp = 3.0 + (x * -12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 0.215]], $MachinePrecision]], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 0.214999999999999997 < x

   1. Initial program 98.9%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 98.9%

         \[\leadsto 3 \cdot \color{blue}{\left(\left(x \cdot 3\right) \cdot x - \left(x \cdot 4 - 1\right)\right)} \]

      2. associate-*l* 98.9%

         \[\leadsto 3 \cdot \left(\color{blue}{x \cdot \left(3 \cdot x\right)} - \left(x \cdot 4 - 1\right)\right) \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \left(3 \cdot x\right) - \left(x \cdot 4 - 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.9%

      \[\leadsto 3 \cdot \color{blue}{\left(1 + \left(-4 \cdot x + 3 \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 98.9%

         \[\leadsto 3 \cdot \left(1 + \color{blue}{\left(3 \cdot {x}^{2} + -4 \cdot x\right)}\right) \]

      2. fma-def 98.9%

         \[\leadsto 3 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(3, {x}^{2}, -4 \cdot x\right)}\right) \]

   7. Simplified 98.9%

      \[\leadsto 3 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(3, {x}^{2}, -4 \cdot x\right)\right)} \]

   8. Taylor expanded in x around inf 97.8%

      \[\leadsto \color{blue}{-12 \cdot x + 9 \cdot {x}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 97.8%

         \[\leadsto -12 \cdot x + 9 \cdot \color{blue}{\left(x \cdot x\right)} \]

      2. associate-*r* 97.8%

         \[\leadsto -12 \cdot x + \color{blue}{\left(9 \cdot x\right) \cdot x} \]

      3. *-commutative 97.8%

         \[\leadsto -12 \cdot x + \color{blue}{\left(x \cdot 9\right)} \cdot x \]

      4. distribute-rgt-out 98.6%

         \[\leadsto \color{blue}{x \cdot \left(-12 + x \cdot 9\right)} \]

   10. Simplified 98.6%

       \[\leadsto \color{blue}{x \cdot \left(-12 + x \cdot 9\right)} \]

   11. Taylor expanded in x around inf 97.7%

       \[\leadsto x \cdot \color{blue}{\left(9 \cdot x\right)} \]
   12. Step-by-step derivation

       1. *-commutative 97.7%

          \[\leadsto x \cdot \color{blue}{\left(x \cdot 9\right)} \]

   13. Simplified 97.7%

       \[\leadsto x \cdot \color{blue}{\left(x \cdot 9\right)} \]

   if -1.55000000000000004 < x < 0.214999999999999997

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto 3 \cdot \color{blue}{\left(\left(x \cdot 3\right) \cdot x - \left(x \cdot 4 - 1\right)\right)} \]

      2. associate-*l* 100.0%

         \[\leadsto 3 \cdot \left(\color{blue}{x \cdot \left(3 \cdot x\right)} - \left(x \cdot 4 - 1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \left(3 \cdot x\right) - \left(x \cdot 4 - 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.4%

      \[\leadsto \color{blue}{3 + -12 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto 3 + \color{blue}{x \cdot -12} \]

   7. Simplified 98.4%

      \[\leadsto \color{blue}{3 + x \cdot -12} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 0.215\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(1 + x \cdot \left(3 \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (+ 1.0 (* x (* 3.0 x)))))

C

double code(double x) {
	return 3.0 * (1.0 + (x * (3.0 * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * (1.0d0 + (x * (3.0d0 * x)))
end function

Java

public static double code(double x) {
	return 3.0 * (1.0 + (x * (3.0 * x)));
}

Python

def code(x):
	return 3.0 * (1.0 + (x * (3.0 * x)))

Julia

function code(x)
	return Float64(3.0 * Float64(1.0 + Float64(x * Float64(3.0 * x))))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * (1.0 + (x * (3.0 * x)));
end

Wolfram

code[x_] := N[(3.0 * N[(1.0 + N[(x * N[(3.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Step-by-step derivation

   1. associate-+l- 99.4%

      \[\leadsto 3 \cdot \color{blue}{\left(\left(x \cdot 3\right) \cdot x - \left(x \cdot 4 - 1\right)\right)} \]

   2. associate-*l* 99.4%

      \[\leadsto 3 \cdot \left(\color{blue}{x \cdot \left(3 \cdot x\right)} - \left(x \cdot 4 - 1\right)\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{3 \cdot \left(x \cdot \left(3 \cdot x\right) - \left(x \cdot 4 - 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.4%

   \[\leadsto 3 \cdot \color{blue}{\left(1 + \left(-4 \cdot x + 3 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 99.4%

      \[\leadsto 3 \cdot \left(1 + \color{blue}{\left(3 \cdot {x}^{2} + -4 \cdot x\right)}\right) \]

   2. fma-def 99.4%

      \[\leadsto 3 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(3, {x}^{2}, -4 \cdot x\right)}\right) \]

7. Simplified 99.4%

   \[\leadsto 3 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(3, {x}^{2}, -4 \cdot x\right)\right)} \]

8. Taylor expanded in x around 0 99.4%

   \[\leadsto 3 \cdot \left(1 + \color{blue}{\left(-4 \cdot x + 3 \cdot {x}^{2}\right)}\right) \]
9. Step-by-step derivation

   1. +-commutative 99.4%

      \[\leadsto 3 \cdot \left(1 + \color{blue}{\left(3 \cdot {x}^{2} + -4 \cdot x\right)}\right) \]

   2. unpow2 99.4%

      \[\leadsto 3 \cdot \left(1 + \left(3 \cdot \color{blue}{\left(x \cdot x\right)} + -4 \cdot x\right)\right) \]

   3. associate-*r* 99.4%

      \[\leadsto 3 \cdot \left(1 + \left(\color{blue}{\left(3 \cdot x\right) \cdot x} + -4 \cdot x\right)\right) \]

   4. distribute-rgt-out 99.8%

      \[\leadsto 3 \cdot \left(1 + \color{blue}{x \cdot \left(3 \cdot x + -4\right)}\right) \]

   5. *-commutative 99.8%

      \[\leadsto 3 \cdot \left(1 + x \cdot \left(\color{blue}{x \cdot 3} + -4\right)\right) \]

10. Simplified 99.8%

    \[\leadsto 3 \cdot \left(1 + \color{blue}{x \cdot \left(x \cdot 3 + -4\right)}\right) \]

11. Taylor expanded in x around inf 97.4%

    \[\leadsto 3 \cdot \left(1 + x \cdot \color{blue}{\left(3 \cdot x\right)}\right) \]
12. Step-by-step derivation

    1. *-commutative 97.4%

       \[\leadsto 3 \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 3\right)}\right) \]

13. Simplified 97.4%

    \[\leadsto 3 \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 3\right)}\right) \]

14. Final simplification 97.4%

    \[\leadsto 3 \cdot \left(1 + x \cdot \left(3 \cdot x\right)\right) \]
15. Add Preprocessing

Alternative 6: 50.6% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 3 \end{array} \]

FPCore

(FPCore (x) :precision binary64 3.0)

C

double code(double x) {
	return 3.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0
end function

Java

public static double code(double x) {
	return 3.0;
}

Python

def code(x):
	return 3.0

Julia

function code(x)
	return 3.0
end

MATLAB

function tmp = code(x)
	tmp = 3.0;
end

Wolfram

code[x_] := 3.0

Derivation

1. Initial program 99.4%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Step-by-step derivation

   1. associate-+l- 99.4%

      \[\leadsto 3 \cdot \color{blue}{\left(\left(x \cdot 3\right) \cdot x - \left(x \cdot 4 - 1\right)\right)} \]

   2. associate-*l* 99.4%

      \[\leadsto 3 \cdot \left(\color{blue}{x \cdot \left(3 \cdot x\right)} - \left(x \cdot 4 - 1\right)\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{3 \cdot \left(x \cdot \left(3 \cdot x\right) - \left(x \cdot 4 - 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 50.3%

   \[\leadsto \color{blue}{3} \]

6. Final simplification 50.3%

   \[\leadsto 3 \]
7. Add Preprocessing

Developer target: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x))))

C

double code(double x) {
	return 3.0 + (((9.0 * x) * x) - (12.0 * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + (((9.0d0 * x) * x) - (12.0d0 * x))
end function

Java

public static double code(double x) {
	return 3.0 + (((9.0 * x) * x) - (12.0 * x));
}

Python

def code(x):
	return 3.0 + (((9.0 * x) * x) - (12.0 * x))

Julia

function code(x)
	return Float64(3.0 + Float64(Float64(Float64(9.0 * x) * x) - Float64(12.0 * x)))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 + (((9.0 * x) * x) - (12.0 * x));
end

Wolfram

code[x_] := N[(3.0 + N[(N[(N[(9.0 * x), $MachinePrecision] * x), $MachinePrecision] - N[(12.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024019 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x)))

  (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))